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Talk:Buchholz's function
Non-unique normal form According to the definition in the article, both ψ0(ψ1(0)) and ψ0(ψ0(ψ1(0))) are in normal form, while they both are equal to ε0. This is obviously wrong, the normal form should be unique. Janek37 (talk) 18:52, May 18, 2019 (UTC) : Right. It was obviously wrong. I replaced it be a correct definition. : p-adic 22:43, May 18, 2019 (UTC) :: Now μk came out of nowhere, I guess it should be νi? :: Janek37 (talk) 07:07, May 19, 2019 (UTC) ::: Oops. Exactly. I fixed it. ::: p-adic 07:59, May 19, 2019 (UTC) Idea for smaller \(\Omega_\alpha\) Instead of taking uncountable ordinals or Church-Kleene numbers, what if we just take Ωα+1 = ψα(Λ) (and of course limit for limits)? It looks like this way all ψ0(α) for α < Λ remain the same, while all other functions now give countable and computable values. The limit of this should be Λ defined as min{β|ψβ(0)=β} in the new notation. This definition is a little circular, but I think it works. Janek37 (talk) 19:16, May 18, 2019 (UTC) : Your definition actually causes a circular logic, and hence you need to estimate \(\Lambda\) in an effective way. For example, you can estimate your \(\Lambda\) by using the original \(\psi\) with uncomputable ordinals or Church-Kleene ordinals. : p-adic 22:54, May 18, 2019 (UTC) :: I was thinking like, the only thing we need from Ωα is to be larger than all values of functions ψβ, β<α. I think it's easy to prove it with my definition and then the circular logic is not that much of a problem, Another approach is Taranovsky's style: define notation and normal forms and then just enumerate normal forms. Ordering is easy once we limit our scope to normal forms. :: Janek37 (talk) 07:07, May 19, 2019 (UTC) ::: > I was thinking like, the only thing we need from Ωα is to be larger than all values of functions ψβ, β<α. I think it's easy to prove it with my definition and then the circular logic is not that much of a problem. ::: Actually you can avoid a circular logic, but I guess that you are missing the point that for an arbitrary ordinal \(\gamma\), \(\psi\_{\beta}(\gamma)\) with \(\beta < \alpha\) heavily depends on the values of \(\psi_{\alpha}\). Avoiding circular logic is absolutely important in mathematics, and hence statements like "this circular logic is not so serious" do not make sense unless you explicitly completely remove the circular logic. Even if you can prove a desired property for "removing a circular logic" based on your "definition", the argument itself is a circular logic. ::: > Another approach is Taranovsky's style: define notation and normal forms and then just enumerate normal forms. Ordering is easy once we limit our scope to normal forms. ::: It is just the same as the ordinal notation system associated to the OCF. By the way, do you know why we need an OCF in order to define an ordinal notation system (i.e. a notation of Taranovsky's style)? The existence of an OCF satisfying a certain compatibilty condition ensures the well-foundedness of the notation. Therefore you need the original OCF in order to ensure your second approach actually works. ::: I guess that you miss the point that you also need define how to determine whether a given expression is of normal form or not. If you just use formal string in a similar way as Taranovsky's ordinal notation, the condition \(\beta_i \in C_{\nu_i}(\beta_i)\) does not make sense at all. You need a primitive recursive interpretation of the criterion of the normal form. (This problem itself is resolved in Buchholz's original paper, though. What I would like to say is that it is not so easy as you might expect.) ::: p-adic 08:15, May 19, 2019 (UTC) :::: Thank you for your answer. I've actually studied math and I don't miss the points you think I miss, but I understand why it looks like I do. It was just a non-formal idea, which is why I didn't bother to formalise it beyond any ambiguity. In particular, I do realize that the criterion of Buchholz normal form is not simple. Janek37 (talk) 00:50, May 20, 2019 (UTC) ::::: I see. Then I am sorry for guessing that you missed the point. Anyway, the idea itself is actually good. ::::: p-adic 02:46, May 20, 2019 (UTC) Another approach to fundamental sequences I have some other idea for fundamental sequences for \(\psi_0(\alpha)\): \(C_0^n(\alpha)\) are obviously all finite, so we can take from each the largest element smaller than \(\psi_0(\alpha)\). The limit of this sequence has to be \(\psi_0(\alpha)\) (for \(\alpha > 0\)). Janek37 (talk) 21:53, May 29, 2019 (UTC) : Yeah, that kind of a formal choice of a recursive system of fundamental sequences is sometimes used (for many other OCFs). For example, hyp cos introduced a recursive system of fundamental sequences for TON, and I introduced a general method to associate OCFs to notation systems with recursive systems of fundamental sequences here. It works well, but it is very difficult for human beings to compute the resulting fundamental seuqences. Therefore we usually try to find another recursive system of fundamental sequences which are easier to compute. : p-adic 22:19, May 29, 2019 (UTC) For more clarity We should add \(\psi_0(\varepsilon_0) = \psi_0(\varepsilon_0+1) = \psi_0(\text{bigger countable ordinal here}) = \varepsilon_0 \) to the examples with explanations, otherwise it's likely for some people to confuse this \(\psi\) with UNOCF. --Nayuta Ito (talk) 01:47, September 29, 2019 (UTC) : I have done, boss. : p-adic 02:25, September 29, 2019 (UTC) ::# "Insert ordinal here" was not supposed to get into the article. ::# I am not your boss. But anyway, thanks. :: --Nayuta Ito (talk) 02:59, September 29, 2019 (UTC) \(\psi_0(\omega_1^{CK})\) Is it equal to \(\varepsilon_0\)? Triakula (talk) 12:42, January 9, 2020 (UTC) : Exactly! : p-adic 12:48, January 9, 2020 (UTC)